Optimal. Leaf size=25 \[ \frac {1}{3} \tanh ^{-1}\left (\frac {3 x+1}{\sqrt {9 x^2+6 x-8}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {621, 206} \[ \frac {1}{3} \tanh ^{-1}\left (\frac {3 x+1}{\sqrt {9 x^2+6 x-8}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-8+6 x+9 x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{36-x^2} \, dx,x,\frac {6+18 x}{\sqrt {-8+6 x+9 x^2}}\right )\\ &=\frac {1}{3} \tanh ^{-1}\left (\frac {1+3 x}{\sqrt {-8+6 x+9 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 24, normalized size = 0.96 \[ \frac {1}{3} \log \left (\sqrt {9 x^2+6 x-8}+3 x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 20, normalized size = 0.80 \[ -\frac {1}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} + 6 \, x - 8} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 21, normalized size = 0.84 \[ -\frac {1}{3} \, \log \left ({\left | -3 \, x + \sqrt {9 \, x^{2} + 6 \, x - 8} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 1.20 \[ \frac {\sqrt {9}\, \ln \left (\frac {\left (9 x +3\right ) \sqrt {9}}{9}+\sqrt {9 x^{2}+6 x -8}\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 22, normalized size = 0.88 \[ \frac {1}{3} \, \log \left (18 \, x + 6 \, \sqrt {9 \, x^{2} + 6 \, x - 8} + 6\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 20, normalized size = 0.80 \[ \frac {\ln \left (3\,x+\sqrt {9\,x^2+6\,x-8}+1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {9 x^{2} + 6 x - 8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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